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We present a new data-driven reduced-order modeling approach to efficientlysolve parametrized partial differential equations (PDEs) for many-queryproblems. This work is inspired by the concept of implicit neuralrepresentation (INR), which models physics signals in a continuous manner andindependent of spatial/temporal discretization. The proposed framework encodesPDE and utilizes a parametrized neural ODE (PNODE) to learn latent dynamicscharacterized by multiple PDE parameters. PNODE can be inferred by ahypernetwork to reduce the potential difficulties in learning PNODE due to acomplex multilayer perceptron (MLP). The framework uses an INR to decode thelatent dynamics and reconstruct accurate PDE solutions. Further, aphysics-informed loss is also introduced to correct the prediction of unseenparameter instances. Incorporating the physics-informed loss also enables themodel to be fine-tuned in an unsupervised manner on unseen PDE parameters. Anumerical experiment is performed on a two-dimensional Burgers equation with alarge variation of PDE parameters. We evaluate the proposed method at a largeReynolds number and obtain up to speedup of O(10^3) and ~1% relative error tothe ground truth values.

Reduced-order modeling for parameterized PDEs via implicit neural representations